theorem
  a,b '||' c,d & a,c '||' b,d & not a,b,c are_collinear implies
   ex x st x,a,d are_collinear
  & x,b,c are_collinear
proof
  assume that
A1: a,b '||' c,d and
A2: a,c '||' b,d and
A3: not a,b,c are_collinear;
  a,b // c,d by A1,A2,A3,Th14;
  then consider x such that
A4: Mid a,x,d and
A5: Mid b,x,c by A3,Th23;
  b,x,c are_collinear by A5,DIRAF:28;
  then
A6: x,b,c are_collinear by DIRAF:30;
  a,x,d are_collinear by A4,DIRAF:28;
  then x,a,d are_collinear by DIRAF:30;
  hence thesis by A6;
end;
